The Internal Direct Product Theorem For Finite Groups

If a group is finite we can modify The Internal Direct Product Theorem as below.

The Internal Direct Product Theorem

Ifandare subgroups of a groupthenis an isomorphism if and only if the following conditions are satisfied:

(ifis finite then).

(Ifis finite thenandare coprime).

andare normal subgroups of

Proof:

so we must prove that all theare distinct. Suppose not, so thatbut thenThe left hand side is inand the right hand side is inso both are equal to e since the intersection is trivial soand All theare distinct therefore andThis implies thatis one to one. Sinceis also onto.

Ifare subgroups of G,is a subgroup of bothandBy Lagrange's Theorem,must divide bothandbut these are coprime sohence